Find he units digit of {2013^1} + {2013^2} + {2013^3} + .... + {2013^{2013}}.

To find the units digit of the sum \(2013^1 + 2013^2 + 2013^3 + \dots + 2013^{2013}\), we focus on the units digit of \(2013^n\) for each \(n\).

### Step 1: Identify the units digit of \(2013^n\)
The units digit of \(2013^n\) is determined by the units digit of \(3^n\) because the other digits do not affect the units digit.

### Step 2: Determine the pattern in the units digits of powers of 3
Let's compute the units digits of the first few powers of 3:
- \(3^1 = 3\) (units digit = 3)
- \(3^2 = 9\) (units digit = 9)
- \(3^3 = 27\) (units digit = 7)
- \(3^4 = 81\) (units digit = 1)
- \(3^5 = 243\) (units digit = 3)

Notice that the units digits repeat every 4 powers: 3, 9, 7, 1.

### Step 3: Apply the pattern to find the units digit of each term
Since the pattern repeats every 4 terms, we calculate the remainder when the exponent is divided by 4 to determine the corresponding units digit:
- If \(n \equiv 1 \ (\text{mod} \ 4)\), the units digit of \(2013^n\) is 3.
- If \(n \equiv 2 \ (\text{mod} \ 4)\), the units digit of \(2013^n\) is 9.
- If \(n \equiv 3 \ (\text{mod} \ 4)\), the units digit of \(2013^n\) is 7.
- If \(n \equiv 0 \ (\text{mod} \ 4)\), the units digit of \(2013^n\) is 1.

### Step 4: Find the sum of the units digits from \(2013^1\) to \(2013^{2013}\)
We now add the units digits for the exponents from 1 to 2013. Since 2013 is not a multiple of 4, we calculate how many complete cycles of 4 occur in the sum and then add the remainder.

- \(2013 \div 4 = 503\) complete cycles, with a remainder of 1.

Each cycle contributes a units digit sum of \(3 + 9 + 7 + 1 = 20\).

Therefore, the sum of the units digits is:
\[
503 \times 20 + 3 = 10060 + 3 = 10063
\]
The units digit of 10063 is **3**.

### Final Answer:
The units digit of the entire sum \(2013^1 + 2013^2 + \dots + 2013^{2013}\) is **3**.

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Would you like further explanation on any part? Here are some related questions you might find interesting:

1. What is the units digit of \(7^{1234}\)?
2. How does modular arithmetic help in finding units digits?
3. What is the units digit of \(2^{50}\)?
4. Can this method be applied to other bases, such as finding the units digit of \(2017^n\)?
5. How would you calculate the units digit of large sums like \(123^1 + 123^2 + \dots + 123^{1000}\)?

**Tip:** When working with powers and units digits, always look for repeating patterns in the sequence to simplify your calculations.

Learn how to find the units digit of the sum 2013^1 + 2013^2 + ... + 2013^2013 using modular arithmetic and patterns in units digits. This problem explores the units digits of powers of 2013 and their cyclic nature, providing a detailed step-by-step solution. Suitable for advanced high school students interested in number theory and modular arithmetic.

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